Qubit Geometry through Holomorphic Quantization

Abstract We develop a wave mechanics formalism for qubit geometry using holomorphic functions and Möbius transformations, providing a geometric perspective on quantum computation. This framework extends the standard Hilbert space description, offering a natural interpretation of standard quantum gates on the Riemann sphere that is examined through their Möbius action on holomorphic wavefunction. These wavefunctions emerge via a quantization process, with the Riemann sphere serving as the classical phase space of the qubit geometry. We quantize this space using Isham’s canonical group quantization with holomorphic polarization, yielding holomorphic wavefunctions and spin angular momentum operators that recover the standard SU (2) algebra with interesting geometric properties. Such properties reveal how geometric transformations induce quantum logic gates on the Riemann sphere, providing a novel perspective on quantum information processing. This result provides a new direction for exploring quantum computation through Isham’s canonical group quantization and its holomorphic polarization method.